Aproksimacije rešenja stohastičkih diferencijalnih jednačina primenom Taylor-ovih redova

Abstract

The subject of the doctoral dissertation is the application of the Taylor formula for the coefficients of various types of stochastic differential equations, for the purpose of the approximation of theirs solutions under non standard conditions, such as the global Lipschitz condition and the linear growth condition. Under certain assumptions, the almost sure convergence and the convergence in the p-th mean, p>0, of the sequence of approximate solutions towards the solution of the initial equation, is shown. The rate of the Lp convergence increases as the orders of the Taylor approximations of the coefficients of the initial equation increase. Shown results are illustrated through the examples which are designed such that the global Lipschitz condition and/or the linear growth condition for the drift and diffusion coefficients are not satisfied. That way, the need for the shown results is satisfied. Techniques used in the proofs are determined by the type of the considered equation, as well as by the conditions which are assumed for the coefficients of the equations.